197
17.2. The Fabri–Picasso theorem
symmetry, this will not be manifested in the form of multiplets of massdegenerate particles.
The preceding italicized sentence does correctly define what is meant by
a spontaneously broken symmetry in field theory, but there is another way of
thinking about it which is somewhat less abstract though also less rigorous.
The basic condition is Q ˆ |0› / = 0, and it seems tempting to infer that, in this
case, the application of Q ˆ to the vacuum gives, not zero, but another possible
′
vacuum, |0› . Thus we have the physically suggestive idea of ‘degenerate
vacua’ (they must be degenerate since [ Q, H
ˆ ] = 0). We shall see in a moment
why this notion, though intuitively helpful, is not rigorous.
It would seem, in any case, that the properties of the vacuum are allimportant, so we begin our discussion with a somewhat formal, but nonetheless fundamental, theorem about the quantum field vacuum.
17.2 The Fabri–Picasso theorem
Suppose that a given Lagrangian L ˆ is invariant under some one-parameter
continuous global internal symmetry with a conserved Noether current ˆ j
μ ,
such that ∂ μ ˆ j
μ = 0. The associated ‘charge’ is the Hermitian operator Q ˆ =
∫ ˆ j
0 d
3
x, and Q ˆ ˙ = 0. We have hitherto assumed that the transformations of
such a U(1) group are representable in the space of physical states by unitary
operations U ˆ (λ) = exp iλQ ˆ for arbitrary λ, with the vacuum invariant under
ˆ
U , so that Q ˆ |0› = 0. Fabri and Picasso (1966) showed that there are actually
two possibilities:
ˆ
(i) Q|0› = 0, and |0› is an eigenstate of Q ˆ with eigenvalue 0, so that
ˆ
|0› is invariant under U ˆ (i.e. U |0› = |0›);
or
ˆ
(ii) Q|0› does not exist in the space (its norm is infinite).
The statement (ii) is technically more correct than the more intuitive stateˆ
ˆ
ments ‘Q|0› = 0’
/
or ‘U |0› = |0›
′ ’, suggested above.
To prove this result, consider the vacuum matrix element ‹0| ˆ j
0 (x)Q ˆ |0›.
From translation invariance, implemented by the unitary operator
2 U ˆ (x) =
exp iP ˆ · x (where P ˆ μ is the 4-momentum operator) we obtain
iP ˆ ·x ˆ
P ·x ˆ
‹0| ˆ j
0 (x)Q ˆ |0› = ‹0|e
j
0 (0)e
−i ˆ Q|0›
iP ˆ ·x ˆ
−iP ˆ
= ‹0|e
j
0 (0) Q ˆ e
·x
|0›
2 If this seems unfamiliar, it may be regarded as the 4-dimensional generalization of the
transformation (I.7) in appendix I of volume 1, from Schr¨ odinger picture operators at t = 0
to Heisenberg operators at t / = 0.
17.2. The Fabri–Picasso theorem
symmetry, this will not be manifested in the form of multiplets of massdegenerate particles.
The preceding italicized sentence does correctly define what is meant by
a spontaneously broken symmetry in field theory, but there is another way of
thinking about it which is somewhat less abstract though also less rigorous.
The basic condition is Q ˆ |0› / = 0, and it seems tempting to infer that, in this
case, the application of Q ˆ to the vacuum gives, not zero, but another possible
′
vacuum, |0› . Thus we have the physically suggestive idea of ‘degenerate
vacua’ (they must be degenerate since [ Q, H
ˆ ] = 0). We shall see in a moment
why this notion, though intuitively helpful, is not rigorous.
It would seem, in any case, that the properties of the vacuum are allimportant, so we begin our discussion with a somewhat formal, but nonetheless fundamental, theorem about the quantum field vacuum.
17.2 The Fabri–Picasso theorem
Suppose that a given Lagrangian L ˆ is invariant under some one-parameter
continuous global internal symmetry with a conserved Noether current ˆ j
μ ,
such that ∂ μ ˆ j
μ = 0. The associated ‘charge’ is the Hermitian operator Q ˆ =
∫ ˆ j
0 d
3
x, and Q ˆ ˙ = 0. We have hitherto assumed that the transformations of
such a U(1) group are representable in the space of physical states by unitary
operations U ˆ (λ) = exp iλQ ˆ for arbitrary λ, with the vacuum invariant under
ˆ
U , so that Q ˆ |0› = 0. Fabri and Picasso (1966) showed that there are actually
two possibilities:
ˆ
(i) Q|0› = 0, and |0› is an eigenstate of Q ˆ with eigenvalue 0, so that
ˆ
|0› is invariant under U ˆ (i.e. U |0› = |0›);
or
ˆ
(ii) Q|0› does not exist in the space (its norm is infinite).
The statement (ii) is technically more correct than the more intuitive stateˆ
ˆ
ments ‘Q|0› = 0’
/
or ‘U |0› = |0›
′ ’, suggested above.
To prove this result, consider the vacuum matrix element ‹0| ˆ j
0 (x)Q ˆ |0›.
From translation invariance, implemented by the unitary operator
2 U ˆ (x) =
exp iP ˆ · x (where P ˆ μ is the 4-momentum operator) we obtain
iP ˆ ·x ˆ
P ·x ˆ
‹0| ˆ j
0 (x)Q ˆ |0› = ‹0|e
j
0 (0)e
−i ˆ Q|0›
iP ˆ ·x ˆ
−iP ˆ
= ‹0|e
j
0 (0) Q ˆ e
·x
|0›
2 If this seems unfamiliar, it may be regarded as the 4-dimensional generalization of the
transformation (I.7) in appendix I of volume 1, from Schr¨ odinger picture operators at t = 0
to Heisenberg operators at t / = 0.